Ginzburg–Landau equations on Riemann surfaces of higher genus

Chouchkov, D.¹ ; Ercolani, N.M.² ; Rayan, S.³ ; Sigal, I.M.¹

¹ Dept. of Math., U. of Toronto, Toronto, ON, M5S 2E4, Canada
² Dept. of Math., U. of Arizona, Tucson, AZ, 85721-0089, USA
³ Dept. of Math. & Stats., U. of Saskatchewan, Saskatoon, SK, S7N 5E6, Canada

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 1, pp. 79-103.

Abstract (VO)

We study the Ginzburg–Landau equations on Riemann surfaces of arbitrary genus. In particular, we

• – construct explicitly the (local moduli space of gauge-equivalent) solutions in the neighborhood of the constant curvature ones;
• – classify holomorphic structures on line bundles arising as solutions to the equations in terms of the degree, the Abel–Jacobi map, and symmetric products of the surface;
• – determine the form of the energy and identify when it is below the energy of the constant curvature (normal) solutions.

Résumé

Nous étudions les équations de Ginzburg–Landau définies sur des surfaces de Riemann de genre arbitraire. En particulier,

• – nous construisons explicitement l'espace des modules locaux des solutions (équivalentes par transformation de jauge) dans le voisinage des solutions de courbure constante ;
• – nous classifions les structures holomorphiques dans les fibrés en droites qui apparaissent comme solutions de ces équations, en fonction de leur degré, de l'application d'Abel–Jacobi, et des produits symétriques de surface ;
• – nous obtenons une expression pour l'énergie et identifions dans quelles conditions elle est inférieure à l'énergie des solutions (normales) de courbure constante.

Détail

DOI : 10.1016/j.anihpc.2019.04.002

Keywords: Ginzburg–Landau equations, Holomorphic bundles, Vortices, Vortex lattices, Superconductivity, Elliptic equations on Riemann surfaces

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